For any countably infinite graph G, Ramsey's theorem guarantees an infinite monochromatic copy of G in any r-coloring of the edges of the countably infinite complete graph K N . Taking this a step further, it is natural to wonder how "large" of a monochromatic copy of G we can find with respect to some measure -for instance, the density (or upper density) of the vertex set of G in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked.In one of the few results in the area, Erdős and Galvin proved that in every 2-coloring of K N , there exists a monochromatic path whose vertex set has upper density at least 2/3, but it is not possible to do better than 8/9. They also showed that for some sequence n → 0, there exists a monochromatic path P such that for infinitely many n, the set {1, 2, ..., n} contains the first ( 1 3+ √ 3 − n )n vertices of P , but it is not possible to do better than 2n/3. We improve both results, in the former case achieving an upper density at least 3/4 and in the latter case obtaining a tight bound of 2/3. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.
Abstract. We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered Ktheory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.
We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.
We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group G is bounded by 2 · h(G)! − 1, where h(G) is the Hirsch length of G. This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.
We prove, using a weakening of the Proper Forcing Axiom, that any homemomorphism betweenČech-Stone remainders of any two locally compact, zero-dimensional Polish spaces is induced by a homeomorphism between their cocompact subspaces.
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